Singularity & Schwartzchild radius asymptote

I was talking with someone about Hawking Radiation and black holes and I realised something that I'd like to fact check.

• all black holes are singularities
• Singularities are an infinitely small point
• The Schwartzschild radius shrinks as the black hole loses energy
• It cannot, however ever reach a truly infinitely small point
• Therefore, the 'event horizon' will never reveal the singularity
• Therefore, black holes get smaller but will never truly disappear

It's like a Xenos paradox: it will draw closer and closer to the center, but cannot truly ever reach it.
Can anyone tell me where I've made the mistakes that I am sure are there?